TORONTO/NEW YORK, Feb 27 (Reuters) - Milestone Apartments Real Estate Investment Trust, which last month agreed to be acquired by Starwood Capital Group for about C$1.7 billion ($1.3 billion), has started talks with the U.S. private investment firm about raising the initial bid, people familiar with the situation told Reuters.

Milestone’s units, which were unchanged before the Reuters report, rose 2 percent to C$21.48 as volume jumped, crossing the current offer of C$21.18.

Milestone and Starwood could agree on a higher price and make an announcement as soon as early this week, the sources said. But they cautioned there was no certainty a deal would be reached at a higher price.

The sources declined to be identified as the talks are confidential. Spokesmen for both Milestone and Starwood declined to comment.

On Jan. 19, Milestone agreed to be bought out by Starwood for $16.15 per Milestone unit in an all-cash transaction.

Based on currency exchange rates at the time, it translated to a premium of about 9.2 percent above the unit’s closing price of C$19.66 before the transaction was announced.

Dallas-based Milestone, which went public in Toronto in 2013, owns and manages apartment properties targeting blue-collar workers across the U.S. southeast and southwest.

With a focus on real estate, Barry Sternlicht-led Starwood Capital manages assets of about $52 billion.

Last week, proxy advisory service Glass Lewis encouraged unitholders to vote for the transaction, while ISS went the other way.

“The fact pattern in the transaction indicates speed and certainty were prioritized over price, apparently out of concerns that cyclical factors will put pressure on REIT valuations,” ISS said in its report on Feb. 22.

In a response the same day, Milestone said its “board and special committee engaged in a comprehensive process to maximize value for the REIT’s unitholders.”

The trust received some approaches in the past two years, Milestone said in a recent regulatory filing, but had not received an alternative bid since the Starwood deal.

In its report, Glass Lewis said “the purchase price represents a compelling value at which Milestone unitholders can cash out their investment in the REIT and immediately realize an assured value, in cash, at a meaningful premium.”

The deadline to vote on the existing bid is March 3. ($1 = 1.3095 Canadian dollars) (Reporting by John Tilak in Toronto and Carl O‘Donnell in New York; Editing by Denny Thomas and Jeffrey Benkoe)